**3. Experimental setup for the radiation intensity distribution study**

The main device to study the X-ray intensity distribution was the HZG-4 diffractometer manufactured by Carl Zeiss Jena Firm. We produced some modification of the device by its detector circle radius increasing up to 500 mm. In the modification result, the measurement space resolution improved in three times. The measurement spectroscopic circuit was completed by NIM standard units produced by Ortec firm. The shaping time of amplifier unit was selected as 0.5 μs. Such selection allowed to get the pulse registration count rate up to 100 kHz. The design of our registration setup is presented in **Figure 7**. X-ray diffractometer used as the setup background is characterized by scanning regimes in nonstop function and start-stop moving with a minimum step of δ(2θ) = 0.001°. X-ray detector was equipped by slit-cut arrester with a width of *s* = 0.1 mm and a height of *h* = 10 mm and Soller slit system limiting the registered flux vertical divergence by value near 2°. X-ray flux take-off angle was selected near 6°. Main volume of experimental investigations was executed by X-ray tube BSW-24 (Cu) in regime *U* = 20 keV and *I* = 10 mA. Similar tube with Fe anode was exploited in some selective measurements. The X-ray space intensity distribution data collection was produced with the use of Cu filter attenuator characterized by the CuKα radiation decreasing factor *K* = 200. For the energy spectrum characterization, our facility setup was equipped by a pulse multichannel analyzer ACCUSPEC Canberra Packard in the form of PC computer board. In measurements of PXW parameters, we used the characteristic part of initial X-ray spectra only.

#### **Figure 7.**

fixed on the waveguide reflector by specific oil (*n* = 1.45). The prism could change its position on the reflector surface. Emergent light beam in the measurement process registered by standard photodiode equipped by circular aperture with a diameter of *d* = 0.5 mm. In the measurements process, the light beam in transit through the prism incidented on the waveguide, underwent the attenuated total internal reflection on the waveguide slit clearance and a lux meter recorded the reflection intensity. The normalization measurement was executed by using the waveguide with a width slit of *s* = 0.12 mm. Specific details of investigations are described elsewhere [34]. The measured data for several waveguides with different

*Experimental data presented the relationship between the waveguide slit width and the reflectivity magnitude*

*are obtained by AITR method. The point characterized the slitless collimator has a specific design.*

*Principle scheme for direct measurements of a waveguide slit width by methods of the attenuated internal total reflection (AITR). I and II are quartz reflectors of a waveguide. Scheme was published, in first, in Ref. [34].*

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

Standard least square method was used for the experimental data fitting allowed to get a relationship between the light beam reflectivity factor and the width of waveguide slit clearance. In process of the slitless collimator study, we registered the gap width variation in interval 0–60 nm at the prism translation along the slitless unit. In result, we concluded that the slitless collimator is characterized by

slit widths are presented in **Figure 6**.

**Figure 5.**

**Figure 6.**

**150**

effective width of the gap *s* = 30 30 nm.

*Instrumental facility for the spatial distribution study of a quasimonochromatic radiation intensity in X-ray beams formed by quartz planar waveguides.*

### **4. Angular radiation intensity distribution in X-ray beams**

In the course of our measurements, the waveguide position in experimental setup in experimental process was not changed. In the experimental process, the distance between the waveguide inlet and the X-ray tube focal position was 75 mm, and the distance between the waveguide outlet and the X-ray detector slit was 460 mm. X-ray flux capture angle calculated on the basis of geometric approach was equal to 0.08° owing to the size of tube focus projection evaluated as 0.1 mm. In experiments, the diffractometer angular step Δ(2θ) was 0.02°. At the same time, the detector slit angular acceptance was 0.01°. The single channel analyzer during experiments transmitted only pulses connected with the Cu characteristic radiation. Scheme of experimental measurement is presented in top position of **Figure 8**. Experimental results are shown in bottom of **Figure 8**.

waveguide reflector surface interaction. It is clear that at any width of the slit clearance, the direct beam will form its own partial peak and will show the linear dependence of its integral intensity variation on the slit clearance width. Its intensity will be equal to zero for the case of the slit clearance zero magnitude. Experimental data showed that the direct beam propagation mechanism was not able to describe the integral beam intensity variation dependence on the slit clearance width, especially, for the nanosize slit interval (a). In this region, the emergent beam total intensity maintains constant magnitude, which is more higher than zero. Calculations showed that the direct beam deposit into the experimental data magnitude for this region is less that 1%. Second region (b) is characterized by the monotonous integral intensity increasing at growth of the slit clearance width. This effect can be connected with a deposit increasing the direct beam in the total X-ray beam intensity. Third region (c) defined as *s* > 3 μm demonstrates sharp intensity growth at the width increasing. This area is characterized by an addition deposit appearing connected with X-ray beam multiple total external reflections on waveguide reflector surfaces. The slit clearance width increasing in this region leads to linear growth of intensity deposits defined by mechanisms of X-ray beam direct and multiple external total reflection propagations. The total integral intensity growth of the waveguide emergent beam break off when the slit clearance width exceeds

*Radiation Fluxes Waveguide-Resonance Phenomenon Discovered in Result of X-Ray Nanosize…*

The experimental data comparison featured for different regions of the slit clearance width and peculiarities discussion of different mechanisms of radiation fluxes propagation insist us on conclusion that the nanosize region (a) is characterized by the specific waveguide-resonance manner of X-ray flux propagation [35]. Devices functioned in frame of the resonance manner we called the planar X-ray waveguide resonators (PXWRs) [29]. PXWR forms the X-ray quasimonochromatic flux as the indivisible ensemble with parameters, which are not depended from the slit clearance width and the initial distribution in radiation flux captured by the device. The waveguide mechanism of the X-ray quasimonochromatic flux propagation featured for the narrow extended slit clearance demonstrates the X-ray radiation density increasing and decreasing the irreversible losses. Angular divergence of PXWR emergent beam is equal to its radiation capture angle, and they cannot exceed twice value of the total reflection critical angle featured for the reflector

The slit clearance width intermediate interval (region b) is connected with two independent deposits into integral intensity defined by direct and quasiresonance beam propagation mechanisms. The increase of slit clearance ensures small growth of the beam integral intensity, but its radiation density diminishes. Spatial intensity distribution featured for this region shows a single-component form owing to small influence of the multiple total external effects on the emergent beam integral intensity. This effect deposit into the intensity becomes decisive factor when the slit clearance width exceeds critical value *s* = 3 μm (c). In this case, emergent beam divergence arrives its maximum Δ*θ* = 2*θ<sup>c</sup>* independently from the device input aperture magnitude. The intensity distribution demonstrates multicomponent

With practical point of view, it is very interesting to compare the radiation density parameter featured for PXWR and waveguides corresponding to "b" and "c" regions with similar parameter featured for X-ray beams formed by a conventional slit-cut device. The beam integral intensity on the slit-cut former output is more higher than one formed by different PXWs. But in the radiation density parameter, the planar extended waveguide structures are more effective. Direct comparison of the slit width is presented in **Figure 9**. Enhanced radiation density peculiar to X-ray beams formed by PXW is connected with width difference of the

size of the radiation source focus projection.

*DOI: http://dx.doi.org/10.5772/intechopen.93174*

material.

structure.

**153**

The slit clearance size interval 0 ≤ *s* ≤ 2 μm was characterized by the absence of the intensity distribution profile transformations for waveguide emergent X-ray beams. Profiles of these distributions were beautifully described by the Gaussian function. It showed some differences in FWHM values and total intensity magnitudes. The intensity magnitude scattering is likely due to our polishing technology wretchedness, but its increasing at *s* > 200 nm is higher as the experimental error. The distribution FWHM was not exceeded the radiation capture angle. At the same time, when the slit clearance width had exceeded 3 μm, the space intensity distribution found new form, which could be interpreted as a set of lines. Experimental data of the waveguide emergent beam total integral intensity dependence on the slit clearance width are presented in **Figure 8**. These dependence can be described by three typical size interval: *s* ≤ 200 nm (a), 200 ≤ *s* ≤ 3000 nm (b), and *s* ≥ 3 μm (c) with different characters of the dependence.

Registered experimental data and common sense allowed to assume that the X-ray waveguide emergent beam consists of some independent deposits. One can suppose by using the geometrical optics concept that one of them is connected with the X-ray beam direct propagation through the waveguide slit clearance without

#### **Figure 8.**

*Experimental scheme for the spatial distribution study of X-ray beam intensity formed by PXWs (1) patterns of CuKαβ intensity spatial distribution in beams formed by PXWs with slit sizes 43 nm, 800 nm, and 35 μm (2), and the experimental dependence of CuKαβ total intensity in beams formed by PXWs on its slit width (3).*

#### *Radiation Fluxes Waveguide-Resonance Phenomenon Discovered in Result of X-Ray Nanosize… DOI: http://dx.doi.org/10.5772/intechopen.93174*

waveguide reflector surface interaction. It is clear that at any width of the slit clearance, the direct beam will form its own partial peak and will show the linear dependence of its integral intensity variation on the slit clearance width. Its intensity will be equal to zero for the case of the slit clearance zero magnitude. Experimental data showed that the direct beam propagation mechanism was not able to describe the integral beam intensity variation dependence on the slit clearance width, especially, for the nanosize slit interval (a). In this region, the emergent beam total intensity maintains constant magnitude, which is more higher than zero. Calculations showed that the direct beam deposit into the experimental data magnitude for this region is less that 1%. Second region (b) is characterized by the monotonous integral intensity increasing at growth of the slit clearance width. This effect can be connected with a deposit increasing the direct beam in the total X-ray beam intensity. Third region (c) defined as *s* > 3 μm demonstrates sharp intensity growth at the width increasing. This area is characterized by an addition deposit appearing connected with X-ray beam multiple total external reflections on waveguide reflector surfaces. The slit clearance width increasing in this region leads to linear growth of intensity deposits defined by mechanisms of X-ray beam direct and multiple external total reflection propagations. The total integral intensity growth of the waveguide emergent beam break off when the slit clearance width exceeds size of the radiation source focus projection.

The experimental data comparison featured for different regions of the slit clearance width and peculiarities discussion of different mechanisms of radiation fluxes propagation insist us on conclusion that the nanosize region (a) is characterized by the specific waveguide-resonance manner of X-ray flux propagation [35]. Devices functioned in frame of the resonance manner we called the planar X-ray waveguide resonators (PXWRs) [29]. PXWR forms the X-ray quasimonochromatic flux as the indivisible ensemble with parameters, which are not depended from the slit clearance width and the initial distribution in radiation flux captured by the device. The waveguide mechanism of the X-ray quasimonochromatic flux propagation featured for the narrow extended slit clearance demonstrates the X-ray radiation density increasing and decreasing the irreversible losses. Angular divergence of PXWR emergent beam is equal to its radiation capture angle, and they cannot exceed twice value of the total reflection critical angle featured for the reflector material.

The slit clearance width intermediate interval (region b) is connected with two independent deposits into integral intensity defined by direct and quasiresonance beam propagation mechanisms. The increase of slit clearance ensures small growth of the beam integral intensity, but its radiation density diminishes. Spatial intensity distribution featured for this region shows a single-component form owing to small influence of the multiple total external effects on the emergent beam integral intensity. This effect deposit into the intensity becomes decisive factor when the slit clearance width exceeds critical value *s* = 3 μm (c). In this case, emergent beam divergence arrives its maximum Δ*θ* = 2*θ<sup>c</sup>* independently from the device input aperture magnitude. The intensity distribution demonstrates multicomponent structure.

With practical point of view, it is very interesting to compare the radiation density parameter featured for PXWR and waveguides corresponding to "b" and "c" regions with similar parameter featured for X-ray beams formed by a conventional slit-cut device. The beam integral intensity on the slit-cut former output is more higher than one formed by different PXWs. But in the radiation density parameter, the planar extended waveguide structures are more effective. Direct comparison of the slit width is presented in **Figure 9**. Enhanced radiation density peculiar to X-ray beams formed by PXW is connected with width difference of the

the detector slit angular acceptance was 0.01°. The single channel analyzer during experiments transmitted only pulses connected with the Cu characteristic radiation. Scheme of experimental measurement is presented in top position of **Figure 8**.

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

The slit clearance size interval 0 ≤ *s* ≤ 2 μm was characterized by the absence of the intensity distribution profile transformations for waveguide emergent X-ray beams. Profiles of these distributions were beautifully described by the Gaussian function. It showed some differences in FWHM values and total intensity magnitudes. The intensity magnitude scattering is likely due to our polishing technology wretchedness, but its increasing at *s* > 200 nm is higher as the experimental error. The distribution FWHM was not exceeded the radiation capture angle. At the same time, when the slit clearance width had exceeded 3 μm, the space intensity distribution found new form, which could be interpreted as a set of lines. Experimental data of the waveguide emergent beam total integral intensity dependence on the slit clearance width are presented in **Figure 8**. These dependence can be described by three typical size interval: *s* ≤ 200 nm (a), 200 ≤ *s* ≤ 3000 nm (b), and *s* ≥ 3 μm (c)

Registered experimental data and common sense allowed to assume that the X-ray waveguide emergent beam consists of some independent deposits. One can suppose by using the geometrical optics concept that one of them is connected with the X-ray beam direct propagation through the waveguide slit clearance without

*Experimental scheme for the spatial distribution study of X-ray beam intensity formed by PXWs (1) patterns of CuKαβ intensity spatial distribution in beams formed by PXWs with slit sizes 43 nm, 800 nm, and 35 μm (2), and the experimental dependence of CuKαβ total intensity in beams formed by PXWs on its slit width (3).*

Experimental results are shown in bottom of **Figure 8**.

with different characters of the dependence.

**Figure 8.**

**152**

with the average wavelength λ0. Owing to this principal limitation, the size of radiation standing wave area in the space over reflector will be bounded. Δλ through the coherence length parameter characterizes the length of an electromagnetic radiation train or the photon longitudinal size. The interference phenomenon is possible if the path difference between the incident and the reflected fluxes does not exceed the magnitude of this parameter. But in any case, the longitudinal size of the interference area cannot exceed of the value. By this, it means that the coherence length of quasimonochromatic radiation is responsible for the longitudinal magnitude evaluation of X-ray standing wave area. In this framework of the phenomenological model, we accepted that the transverse size of the area is approximately equal to the longitudinal one. This premise is absolutely right so far as the real interference effect is connected with the spatial coherence of the quasimono-

*Radiation Fluxes Waveguide-Resonance Phenomenon Discovered in Result of X-Ray Nanosize…*

The next model postulate says that the description of the total X-ray reflection phenomenon must take into account the fundamental principle of a field continuity [37]. According to this principle, the interference field of X-ray standing wave cannot abruptly terminate on the material-vacuum (air) interface. The conventional model of X-ray beam total external reflection suggests that the radiation electromagnetic field amplitude undergoes exponential attenuation in the reflector material [38]. But the principle of electromagnetic field continually demands the exponential low multiplication on the interference term. Visualization of the modification is presented in **Figure 10b** and can be defined by the expression [39]:

> 2*θ <sup>θ</sup>* <sup>þ</sup> *<sup>a</sup>* <sup>þ</sup> *ib <sup>e</sup>*

� �*<sup>e</sup>*

where *θ* is the incident angle; *λ*<sup>0</sup> and *ω*<sup>0</sup> are the wavelength and the angular frequency of the radiation, respectively; *px* is the *x*-component of the photon momentum; and *a* and *b* are presented by the specific expressions [38]:

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>θ</sup>*<sup>2</sup> � <sup>2</sup>*<sup>δ</sup>* � �<sup>2</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>θ</sup>*<sup>2</sup> � <sup>2</sup>*<sup>δ</sup>* � �<sup>2</sup>

<sup>þ</sup> <sup>4</sup>*β*<sup>2</sup>

<sup>þ</sup> <sup>4</sup>*β*<sup>2</sup>

� *<sup>θ</sup>*<sup>2</sup> � <sup>2</sup>*<sup>δ</sup>* � � � �

where *δ* and *β* factors are the formal parameters incoming into the conventional

The *δ*-factor is connected with the volume material polarization effect, and *β*-factor characterizes the attenuation degree of X-ray radiation flux in the material. But we would like to notice that the refractive index introduction in form presented by expression (3) is not correct, in principle. It suggests that the X-ray beam propagation velocity in material volume is higher than the velocity of the electromagnetic wave propagation in vacuum. Professor L.D. Landau at first

**Figure 10a** and **b** displays the principle model for an electromagnetic field distribution in the reflection area over and under the interface. The size of the interference area appeared over the interface is limited by the coherence radiation condition. But the interference area size under the interface is not limited. The entire volume of the reflector will be excited as a result of a flux total reflection on

� *<sup>θ</sup>*<sup>2</sup> � <sup>2</sup>*<sup>δ</sup>* � � � �

2*πiz a*ð Þ þ*ib λ*0

*<sup>i</sup> <sup>ω</sup>*0*t*�2*πpx* ð Þ*<sup>x</sup> E*

!

9 >>>=

>>>;

*n* ¼ 1 � *δ* � *iβ* (3)

<sup>0</sup> (1)

(2)

chromatic flux radiation [36].

*DOI: http://dx.doi.org/10.5772/intechopen.93174*

*E* !

*<sup>T</sup>*ð Þ¼ *z*, *x*, *t*

*<sup>a</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

*<sup>b</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

expression for the material refractive index [38]:

pointed on this collision [40].

its local spot.

**155**

q

q

#### **Figure 9.**

*Experimental dependences of CuKαβ flux radiation densities on slit width for X-ray beams formed by PXWs (a) and by the single slit-cut system (b).*

waveguide slit clearance and the radiation source focus projection. Waveguides capture X-ray radiation fluxes in the angular aperture Δφ ≤ 2θ<sup>c</sup> from radiation source focus projection with width *P* 0.1 mm into the slit clearance with more smaller width. In result, waveguide devices concentrate radiation. According to the data presented in **Figure 9**, PXWR is able to increase X-ray radiation density on three orders in its emergent beam in comparison with beams formed by slit-cut system. It is interesting that the maximum radiation density is expected for the slitless collimator. But the practical use of X-ray slitless system is troubled owing to the absence of the intensity stationary in its emergent beams.

### **5. Waveguide-resonance model for X-ray flux propagation**

**Figure 10a** presents the idealizing scheme of X-ray flux total external reflection, which takes into account the degree of a radiation monochromatization Δλ along

#### **Figure 10.**

*Scheme of the interference field of X-ray standing wave arising from the external total reflection phenomenon in case of the quasimonochromatic flux interaction with material interface (a). Δ*w *is the source focus projection,* R *is the distance between X-ray source and target position, φ is the angular flux divergence, Δ*z *is the penetration depth, and* D *is the standing wave period. Standing wave intensities in the air area and in the reflector volume (b).*

*Radiation Fluxes Waveguide-Resonance Phenomenon Discovered in Result of X-Ray Nanosize… DOI: http://dx.doi.org/10.5772/intechopen.93174*

with the average wavelength λ0. Owing to this principal limitation, the size of radiation standing wave area in the space over reflector will be bounded. Δλ through the coherence length parameter characterizes the length of an electromagnetic radiation train or the photon longitudinal size. The interference phenomenon is possible if the path difference between the incident and the reflected fluxes does not exceed the magnitude of this parameter. But in any case, the longitudinal size of the interference area cannot exceed of the value. By this, it means that the coherence length of quasimonochromatic radiation is responsible for the longitudinal magnitude evaluation of X-ray standing wave area. In this framework of the phenomenological model, we accepted that the transverse size of the area is approximately equal to the longitudinal one. This premise is absolutely right so far as the real interference effect is connected with the spatial coherence of the quasimonochromatic flux radiation [36].

The next model postulate says that the description of the total X-ray reflection phenomenon must take into account the fundamental principle of a field continuity [37]. According to this principle, the interference field of X-ray standing wave cannot abruptly terminate on the material-vacuum (air) interface. The conventional model of X-ray beam total external reflection suggests that the radiation electromagnetic field amplitude undergoes exponential attenuation in the reflector material [38]. But the principle of electromagnetic field continually demands the exponential low multiplication on the interference term. Visualization of the modification is presented in **Figure 10b** and can be defined by the expression [39]:

$$\overrightarrow{E}\_T(\mathbf{z}, \mathbf{x}, t) = \left[ \frac{2\theta}{\theta + a + ib} e^{\frac{2\pi i (a+ib)}{\lambda\_0}} \right] e^{i \left(a \circ t - 2\pi p\_x x\right)} \overrightarrow{E}\_0 \tag{1}$$

where *θ* is the incident angle; *λ*<sup>0</sup> and *ω*<sup>0</sup> are the wavelength and the angular frequency of the radiation, respectively; *px* is the *x*-component of the photon momentum; and *a* and *b* are presented by the specific expressions [38]:

$$\begin{aligned} a^2 &= \frac{1}{2} \left[ \sqrt{\left(\theta^2 - 2\delta\right)^2 + 4\theta^2} - \left(\theta^2 - 2\delta\right) \right] \\\ b^2 &= \frac{1}{2} \left[ \sqrt{\left(\theta^2 - 2\delta\right)^2 + 4\theta^2} - \left(\theta^2 - 2\delta\right) \right] \end{aligned} \tag{2}$$

where *δ* and *β* factors are the formal parameters incoming into the conventional expression for the material refractive index [38]:

$$n = 1 - \delta - i\beta\tag{3}$$

The *δ*-factor is connected with the volume material polarization effect, and *β*-factor characterizes the attenuation degree of X-ray radiation flux in the material. But we would like to notice that the refractive index introduction in form presented by expression (3) is not correct, in principle. It suggests that the X-ray beam propagation velocity in material volume is higher than the velocity of the electromagnetic wave propagation in vacuum. Professor L.D. Landau at first pointed on this collision [40].

**Figure 10a** and **b** displays the principle model for an electromagnetic field distribution in the reflection area over and under the interface. The size of the interference area appeared over the interface is limited by the coherence radiation condition. But the interference area size under the interface is not limited. The entire volume of the reflector will be excited as a result of a flux total reflection on its local spot.

waveguide slit clearance and the radiation source focus projection. Waveguides capture X-ray radiation fluxes in the angular aperture Δφ ≤ 2θ<sup>c</sup> from radiation source focus projection with width *P* 0.1 mm into the slit clearance with more smaller width. In result, waveguide devices concentrate radiation. According to the data presented in **Figure 9**, PXWR is able to increase X-ray radiation density on three orders in its emergent beam in comparison with beams formed by slit-cut system. It is interesting that the maximum radiation density is expected for the slitless collimator. But the practical use of X-ray slitless system is troubled owing to

*Experimental dependences of CuKαβ flux radiation densities on slit width for X-ray beams formed by PXWs*

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

the absence of the intensity stationary in its emergent beams.

**Figure 9.**

**Figure 10.**

**154**

*(a) and by the single slit-cut system (b).*

**5. Waveguide-resonance model for X-ray flux propagation**

**Figure 10a** presents the idealizing scheme of X-ray flux total external reflection, which takes into account the degree of a radiation monochromatization Δλ along

*Scheme of the interference field of X-ray standing wave arising from the external total reflection phenomenon in case of the quasimonochromatic flux interaction with material interface (a). Δ*w *is the source focus projection,* R *is the distance between X-ray source and target position, φ is the angular flux divergence, Δ*z *is the penetration depth, and* D *is the standing wave period. Standing wave intensities in the air area and in the reflector volume (b).*

The external total reflection phenomenon is accompanied by an additional phase shift Δψ [41]. This parameter is the function of the flux incident angle φ. At the critical total reflection angle (φ ffi θc), the additional phase shift strives to zero, but at the sliding incidence (φ ffi 0), Δψ value approximates to "π." The variation of the additional phase shift magnitude influences on the interference area size. Therefore, in framework of the waveguide-resonance model, the solution was accepted that the size of X-ray standing wave interference area is approximately conformed to half magnitude of the coherence length for the radiation flux undergoing the total reflection on the material interface [39].

But in common case, the radiation flux can incident on the PXWR inlet at off-axis conditions. Measures in conditions of an initial X-ray flux off-axis incidence allow to differ the discrete mode structure from one with the continuous character, if the flux divergence is not great. The waveguide-resonance concept predicts that the off-axis incidence of X-ray flux will lead to the appearance of the emergent beam in the form of the double peak for a radiation spatial distribution (**Figure 11b**). The angular distance between the maximum of peaks must be equal to a double

*Radiation Fluxes Waveguide-Resonance Phenomenon Discovered in Result of X-Ray Nanosize…*

magnitude of the incidence angle. It is expected that the intensity of peaks must be approximately equivalent, and its divergences will be correlated with the radiation capture angle. Moreover, the integral intensity of the double-peak structure must be the monotonous function on the incidence angle for the angular interval –θ<sup>c</sup> ≤ θ ≤ θc. The reliable confirmation of all predictions following from the model of X-ray flux waveguide-resonance propagation was obtained in the course of our experimental

The integral intensity of PXWR emergent beam is insignificantly differed from the intensity of X-ray initial beam. Its magnitude can be described by the

*W x*ð Þ¼ *W*0*e*

where *W*<sup>0</sup> is the initial beam intensity, *μ* is the linear absorption factor of reflector material for the radiation transported by PXWR, and *α* is the composite function defined by set factors of physical and geometrical nature. α magnitude is very small, and the attenuation of X-ray flux transported by PXWR is not significant. The experimental measurements showed that the attenuation factor for CuKα radiation transported by the quartz PXWR with a length of *l* � 100 mm can be near some per cents. The intensity losses for PXWR are smaller as the calculated values characterized for the multiple total reflection mechanisms. High radiation transportation efficiency of the planar X-ray waveguide-resonator is the result of the mode structure continuity for the flux propagating through its narrow extended slit

It is very important to notice that X-ray flux transportation by the waveguideresonance mechanism is the result of the spatial coherence of quasimonochromatic radiation irradiated by X-ray tube. Owing to the fundamental physical reasons, a single X-ray photon in conditions of the total external reflection on the material interface cannot undergo interference with itself. The reflection process for X-ray and for other nature waves is accompanied by the Goos-Hanchen wave front displacement of the beam reflection position about the point of the beam incoming

> 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> *<sup>c</sup>* � *<sup>θ</sup>*<sup>2</sup> � � <sup>þ</sup> <sup>2</sup>*i<sup>β</sup>*

where *θ<sup>c</sup>* is the magnitude of the total reflection critical angle. Minimum and maximum magnitudes of the displacement are arrived at *θ* = 0 and *θ* = *θc*, corre-

; <sup>Δ</sup>*x*max <sup>¼</sup> *<sup>λ</sup>*<sup>0</sup>

Calculation of these factors shows that its magnitudes do not exceed the radia-

*πθ<sup>c</sup>*

1 ffiffiffiffiffi

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *θ*2 *<sup>c</sup>* þ 2*iβ*

<sup>q</sup> (5)

<sup>2</sup>*<sup>β</sup>* <sup>p</sup> (6)

*θ*2

*πθ*<sup>2</sup> *c*

q

place, which is presented by the expression [42–44]:

<sup>Δ</sup>*<sup>x</sup>* <sup>¼</sup> *<sup>λ</sup>*<sup>0</sup> *π*

spondingly. The expressions for these values have forms:

tion coherence length and interference takes place.

<sup>Δ</sup>*x*min <sup>¼</sup> *<sup>λ</sup>*<sup>0</sup>

�*αμ<sup>x</sup>* (4)

investigations [36].

*DOI: http://dx.doi.org/10.5772/intechopen.93174*

expression [39]:

clearance.

**157**

If we place two planar dielectric polished reflectors on some distance, we can get the air planar extended slit clearance, which can be used for the realization of X-ray flux multiple external total reflection effects (**Figure 11a**). The consecutive multiple external total reflection phenomena are characterized by appearing the local interference area set. Since every elementary act of X-ray flux total reflection excites material volume of reflector, the next second reflection in the slit clearance on the reflector surface will lead to the material volume excitation. One can find specific X-ray flux incident angles, which will show the phasing of consecutive total reflection on every reflector (magic angles). Peculiarities of X-ray beam propagation through the air slit clearance are depended from the existence or absence of the phasing. The presence of phasing effect allows to transport the X-ray quasimonochromatic flux by PXW with small attenuation. The magic angle existence defines the discrete mode structure featured for the multiple total external reflection mechanisms. This picture is inherent for the X-ray polycapillary optics.

The mechanism of X-ray flux multiple total reflections is very efficient for the description of its propagation through planar wide slit clearances. But this mechanism is not able to characterize peculiarities of the radiation flux transportation by the super narrow planar extended slits. X-ray flux propagation through similar slits can be described on the basis of waveguide-resonance idea.

The conception of X-ray flux waveguide-resonance propagation is accompanied by appearing the X-ray standing wave uniform interference field in all narrow extended slit clearance spaces owing to the mutual overlap of local interference areas (**Figure 11b**). Overlay of these areas will be realized for any magnitudes of incident angles when it does not exceed the value of total reflection critical angle θ<sup>c</sup> for the reflector material. The mode structure conception for PXWR is not existed. Moreover, we can confirm that the radiation coherence length magnitude is the critical parameter for the X-ray flux mechanism propagation change from the multiple external total reflections to the waveguide-resonance proceeding.

The narrow extended slit clearance radiation transport properties discussed above were investigated in the geometry when the projection of X-ray source focus was deposited in the symmetry plane of PXWR [39]. In this measurement geometry, the axis of X-ray incidence flux coincides with the axis of waveguide resonator.

#### **Figure 11.**

*Visualizating schemes of X-ray flux propagation through the planar extended slit clearance by the multiple total reflection mechanisms (a) and by the waveguide-resonance one (b).* P *is the parameter of the interference field protrusion from PXWR slit.*

*Radiation Fluxes Waveguide-Resonance Phenomenon Discovered in Result of X-Ray Nanosize… DOI: http://dx.doi.org/10.5772/intechopen.93174*

But in common case, the radiation flux can incident on the PXWR inlet at off-axis conditions. Measures in conditions of an initial X-ray flux off-axis incidence allow to differ the discrete mode structure from one with the continuous character, if the flux divergence is not great. The waveguide-resonance concept predicts that the off-axis incidence of X-ray flux will lead to the appearance of the emergent beam in the form of the double peak for a radiation spatial distribution (**Figure 11b**). The angular distance between the maximum of peaks must be equal to a double magnitude of the incidence angle. It is expected that the intensity of peaks must be approximately equivalent, and its divergences will be correlated with the radiation capture angle. Moreover, the integral intensity of the double-peak structure must be the monotonous function on the incidence angle for the angular interval –θ<sup>c</sup> ≤ θ ≤ θc. The reliable confirmation of all predictions following from the model of X-ray flux waveguide-resonance propagation was obtained in the course of our experimental investigations [36].

The integral intensity of PXWR emergent beam is insignificantly differed from the intensity of X-ray initial beam. Its magnitude can be described by the expression [39]:

$$\mathcal{W}(\mathbf{x}) = \mathcal{W}\_0 e^{-a\mu\mathbf{x}} \tag{4}$$

where *W*<sup>0</sup> is the initial beam intensity, *μ* is the linear absorption factor of reflector material for the radiation transported by PXWR, and *α* is the composite function defined by set factors of physical and geometrical nature. α magnitude is very small, and the attenuation of X-ray flux transported by PXWR is not significant. The experimental measurements showed that the attenuation factor for CuKα radiation transported by the quartz PXWR with a length of *l* � 100 mm can be near some per cents. The intensity losses for PXWR are smaller as the calculated values characterized for the multiple total reflection mechanisms. High radiation transportation efficiency of the planar X-ray waveguide-resonator is the result of the mode structure continuity for the flux propagating through its narrow extended slit clearance.

It is very important to notice that X-ray flux transportation by the waveguideresonance mechanism is the result of the spatial coherence of quasimonochromatic radiation irradiated by X-ray tube. Owing to the fundamental physical reasons, a single X-ray photon in conditions of the total external reflection on the material interface cannot undergo interference with itself. The reflection process for X-ray and for other nature waves is accompanied by the Goos-Hanchen wave front displacement of the beam reflection position about the point of the beam incoming place, which is presented by the expression [42–44]:

$$
\Delta \mathbf{x} = \frac{\lambda\_0}{\pi} \frac{1}{\sqrt{\left(\theta\_\varepsilon^2 - \theta^2\right) + 2i\beta}} \frac{1}{\sqrt{\theta\_\varepsilon^2 + 2i\beta}} \tag{5}
$$

where *θ<sup>c</sup>* is the magnitude of the total reflection critical angle. Minimum and maximum magnitudes of the displacement are arrived at *θ* = 0 and *θ* = *θc*, correspondingly. The expressions for these values have forms:

$$
\Delta \mathbf{x}\_{\rm min} = \frac{\lambda\_0}{\pi \theta\_\varepsilon^2}; \ \Delta \mathbf{x}\_{\rm max} = \frac{\lambda\_0}{\pi \theta\_\varepsilon} \frac{1}{\sqrt{2\beta}} \tag{6}
$$

Calculation of these factors shows that its magnitudes do not exceed the radiation coherence length and interference takes place.

The external total reflection phenomenon is accompanied by an additional phase shift Δψ [41]. This parameter is the function of the flux incident angle φ. At the critical total reflection angle (φ ffi θc), the additional phase shift strives to zero, but at the sliding incidence (φ ffi 0), Δψ value approximates to "π." The variation of the additional phase shift magnitude influences on the interference area size. Therefore, in framework of the waveguide-resonance model, the solution was accepted that the size of X-ray standing wave interference area is approximately conformed to half magnitude of the coherence length for the radiation flux undergoing the total

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

If we place two planar dielectric polished reflectors on some distance, we can get the air planar extended slit clearance, which can be used for the realization of X-ray flux multiple external total reflection effects (**Figure 11a**). The consecutive multiple external total reflection phenomena are characterized by appearing the local interference area set. Since every elementary act of X-ray flux total reflection excites material volume of reflector, the next second reflection in the slit clearance on the reflector surface will lead to the material volume excitation. One can find specific X-ray flux incident angles, which will show the phasing of consecutive total reflection on every reflector (magic angles). Peculiarities of X-ray beam propagation through the air slit clearance are depended from the existence or absence of the phasing. The presence of phasing effect allows to transport the X-ray quasimonochromatic flux by PXW with small attenuation. The magic angle existence defines the discrete mode structure featured for the multiple total external reflection mechanisms. This picture is inherent for the X-ray polycapillary optics.

The mechanism of X-ray flux multiple total reflections is very efficient for the description of its propagation through planar wide slit clearances. But this mechanism is not able to characterize peculiarities of the radiation flux transportation by the super narrow planar extended slits. X-ray flux propagation through similar slits

The conception of X-ray flux waveguide-resonance propagation is accompanied

by appearing the X-ray standing wave uniform interference field in all narrow extended slit clearance spaces owing to the mutual overlap of local interference areas (**Figure 11b**). Overlay of these areas will be realized for any magnitudes of incident angles when it does not exceed the value of total reflection critical angle θ<sup>c</sup> for the reflector material. The mode structure conception for PXWR is not existed. Moreover, we can confirm that the radiation coherence length magnitude is the critical parameter for the X-ray flux mechanism propagation change from the multiple external total reflections to the waveguide-resonance proceeding.

The narrow extended slit clearance radiation transport properties discussed above were investigated in the geometry when the projection of X-ray source focus was deposited in the symmetry plane of PXWR [39]. In this measurement geometry, the axis of X-ray incidence flux coincides with the axis of waveguide resonator.

*Visualizating schemes of X-ray flux propagation through the planar extended slit clearance by the multiple total reflection mechanisms (a) and by the waveguide-resonance one (b).* P *is the parameter of the interference*

can be described on the basis of waveguide-resonance idea.

**Figure 11.**

**156**

*field protrusion from PXWR slit.*

reflection on the material interface [39].

radiation generated by X-ray tube is smaller than this parameter featured for X-ray quasimonochromatic lines. **Figure 13** shows that the white component intensity falls down approximately two times in all spectral ranges investigated in the experiments. Thus, a planar X-ray waveguide resonator cannot be considered as a restrictive filter for the hard white radiation. But PXWR application for X-ray beam formation decreases the white radiation deposit in the total beam intensity. This effect will be greatest for the smallest slit clearance width. The specific feature of PXWR is the impossibility to use it for β-filtration of X-ray tube initial radiation. The β-filtration procedure for X-ray diffractometry is well known [47]. This procedure is based on the use of the thin film absorber manufactured from the material, which is characterized by the energy absorption edge intervened between *EK*<sup>α</sup> and *EK*<sup>β</sup> of the tube characteristic radiation. Similar β-filter can be built on the basis of planar monocapillar prepared by using the dielectric reflectors containing a significant concentration of atoms characterized by a suitable value of the energy absorption edge. Our direct experiments showed that the similar approach is not right for PXWR. β-Radiation flux excites the uniform interference field of X-ray standing wave in all space of PXWR air slit clearance, and the intensity attenuation

*Radiation Fluxes Waveguide-Resonance Phenomenon Discovered in Result of X-Ray Nanosize…*

*DOI: http://dx.doi.org/10.5772/intechopen.93174*

Specific properties of PXWR are not exhausted by the peculiarities discussed above. For example, the beam formed by the waveguide resonator has the nanosize width and the enhanced radiation density. The beam is not accompanied by diffraction satellites and can be modulated by an external influence. But the planar X-ray waveguide resonator is characterized by two serious lacks. The angular divergence of the beam formed by PXWR of the simplest design is usually near 0.1°, and its real integral intensity is smaller than the integral intensities of beams formed by the slit-cut systems and the polycapillary optic devices on 1–2 orders [39]. The angular divergence of PXWR emergent beam can be decreased without influence on its integral intensity by application of PXWR with specific design, which has name as the composite planar X-ray waveguide resonator (CPXWR) [48].

**Figure 14** presents the results of comparative investigations of X-ray characteristic beam formation by PXWR with the simplest construction (a) and CPXWR (b). Left part of the figure presents the measurement schemes. Spatial distributions of X-ray intensities in beams formed by these devices are shown in the right part of the figure. Radiation capture angle is the same and is equal to Δφ<sup>1</sup> = 0.11°. Composite

*Experimental diffraction patterns for SiO2 (101) monocrystal specimen collected in conditions of a standard Bragg-Brentano geometry (a) and a waveguide-resonator application for the initial beam formation (b). The pattern normalization was carried out on the basis of equivalence of characteristic line intensities. Pattern (a) was registered at BSW-24 (Fe) X-ray tube regime* U *= 25 keV,* I *= 3 mA and pattern (b)* U *= 25 keV,* I *= 9 mA. Geometrical sizes at the measurements were (a)* l*<sup>0</sup> = 235 mm,* l*<sup>1</sup> = 50 mm,* l*<sup>2</sup> = 235 mm,*

S*<sup>1</sup> =* S*<sup>2</sup> = 0.1 mm and (b)* l*<sup>0</sup> = 235 mm,* l*<sup>1</sup> = 50 mm,* l*<sup>2</sup> = 155 mm,* l*<sup>3</sup> = 85 mm,* S*<sup>1</sup> = 0.1 mm,* s*PXWR = 0.1 μm.*

*The collection was carried out without a pulse discrimination.*

is not observed.

**Figure 13.**

**159**
